Recovering initial values from light cone traces of solutions of the wave equation

Abstract

We consider the problem of recovering the initial value, from the trace on the light cone, of the solution of an initial value problem for the wave equation. When the space is odd dimensional, we show that the map from the initial value to the traces of the (even or odd in time) solutions on the light cone is an isometry and we characterize the range of this map and construct its inverse. We do this by relating the problem to the recovery of a function from its spherical means over all spheres through the origin, which in turn is related to the Radon transform inversion via the inversion map on R^n.

This is a revision of an earlier version. The title has been changed (at the request of a referee), constants have been corrected in the formulas in Theorem 3, the formula in Theorem 4 has been corrected and there are corresponding minor changes in the proofs. Sections 1.2 and 1.3 have been swapped and the abstract is a little longer